Triangle upper J upper K upper L is divided into triangle upper P upper K upper Q and quadrilateral upper J upper P upper Q upper L by vertical line upper P upper Q. Point upper P is on line upper J upper K. Point upper Q is on diagonal upper L upper K.

Question

△JKL∼△PKQ , and KP¯¯¯¯¯¯¯¯=14 m, JK¯¯¯¯¯¯¯¯=56 m, and KQ¯¯¯¯¯¯¯¯=21 m. What is the length of KL¯¯¯¯¯¯¯¯ ?

(1 point)
Responses

84 m
84 m

34 m
34 m

5.25 m
5.25 m

72 m

Answer 1

Given that triangles \( \triangle JKL \) and \( \triangle PKQ \) are similar (denoted as \( \triangle JKL \sim \triangle PKQ \)), we can use the properties of similar triangles to find the length of segment \( KL \).

Let's denote:

• \( JK = 56 \) m
• \( KP = 14 \) m
• \( KQ = 21 \) m
• Let \( KL = x \)

Since the two triangles are similar, the corresponding sides are proportional:

\[ \frac{JK}{KP} = \frac{KL}{KQ} \]

Substituting the known lengths into the ratio gives:

\[ \frac{56}{14} = \frac{x}{21} \]

Calculating the left side:

\[ \frac{56}{14} = 4 \]

Thus, we have:

\[ 4 = \frac{x}{21} \]

To solve for \( x \), we can cross-multiply:

\[ 4 \cdot 21 = x \]

\[ 84 = x \]

Thus, the length of \( KL \) is:

\[ KL = 84 \text{ m} \]

Therefore, the correct answer is \( \text{84 m} \).

Answer 2

Triangle upper Y upper X upper Z is divided into triangle upper Y upper B upper A and quadrilateral upper A upper B upper Z upper X by horizontal line upper A upper B.
If △XYZ∼△AYB , and XY¯¯¯¯¯¯¯¯=24 yards, YZ¯¯¯¯¯¯¯=30 yards, and YA¯¯¯¯¯¯¯¯=8 yards, then what is the length of YB¯¯¯¯¯¯¯¯ ?

(1 point)
Responses

12 yards
12 yards

15 yards
15 yards

10 yards
10 yards

25 yards
25 yards

Answer 3

Given that triangles \( \triangle XYZ \) and \( \triangle AYB \) are similar (denoted as \( \triangle XYZ \sim \triangle AYB \)), we can use the properties of similar triangles to find the length of segment \( YB \).

Let's denote:

• \( XY = 24 \) yards
• \( YZ = 30 \) yards
• \( YA = 8 \) yards
• Let \( YB = x \)

Since the two triangles are similar, the corresponding sides are proportional:

\[ \frac{XY}{YA} = \frac{YZ}{YB} \]

Substituting the known lengths into the ratio gives:

\[ \frac{24}{8} = \frac{30}{x} \]

Calculating the left side:

\[ \frac{24}{8} = 3 \]

Now we have:

\[ 3 = \frac{30}{x} \]

To solve for \( x \), we can cross-multiply:

\[ 3x = 30 \]

Dividing both sides by 3 gives:

\[ x = 10 \]

Thus, the length of \( YB \) is:

\[ YB = 10 \text{ yards} \]

Therefore, the correct answer is 10 yards.

Answer 4

Myk is 5 feet and 2 inches tall, and their shadow is 84 inches tall. The flagpole next to them has a shadow that is 31 feet long. Myk uses similar triangles to find the height of the flagpole. Can you calculate the height of the flagpole to the nearest foot?(1 point)
Responses

17 feet
17 feet

23 inches
23 inches

374 feet
374 feet

23 feet
23 feet

Answer 5

To find the height of the flagpole using similar triangles, we can set up a proportion based on the heights and the lengths of the shadows.

First, we need to convert Myk's height from feet and inches to inches. Myk is 5 feet 2 inches tall:

\[ \text{Myk's height in inches} = (5 \times 12) + 2 = 60 + 2 = 62 \text{ inches} \]

Let:

• \( h \) = height of the flagpole (in inches)
• Myk's height = 62 inches
• Myk's shadow = 84 inches
• Flagpole's shadow = 31 feet = \( 31 \times 12 = 372 \) inches

Now we can set up the proportion:

\[ \frac{\text{Myk's height}}{\text{Myk's shadow}} = \frac{\text{Flagpole's height}}{\text{Flagpole's shadow}} \]

Substituting the values we know:

\[ \frac{62}{84} = \frac{h}{372} \]

Cross-multiplying to solve for \( h \):

\[ 62 \times 372 = 84 \times h \]

Calculating the left side:

\[ 62 \times 372 = 23064 \]

So the equation becomes:

\[ 23064 = 84h \]

Now, divide both sides by 84 to find \( h \):

\[ h = \frac{23064}{84} \approx 274 \]

Now we need to convert this height from inches to feet. Since there are 12 inches in a foot:

\[ \text{Height in feet} = \frac{274}{12} \approx 22.83 \]

Rounding to the nearest foot, we get approximately 23 feet.

Thus, the height of the flagpole is 23 feet.